I think that one of the main reasons why the US has the complicated welfare state that it does, instead of simple cash transfers, is that unconditional cash transfers to the poor are unpopular among the general public. They make a lot of people feel like they are being taken advantage of. Instead of working like we are, these people are just mooching off of us. The government is taking our hard-earned money, which we deserve, and giving it to people who don't deserve it. That feeling was one of the main motivations behind welfare reform in the 1990s (recall "welfare queens"). When the word "redistribution" does come up in American politics, it's almost always as an accusation by the right against the left (like in response to Barack Obama's "spread the wealth around" comments during the 2008 election).

If you look at the various safety net programs that the US has (alternate sources), they're designed in a way that avoids that impression that they are just giving your money to the undeserving poor. The recipient seems deserving, because the program does some combination of the following:

• help people who are especially needy, sympathetic, or helpless on their own, like children or the disabled (e.g., SSI, school lunches, CHIP). They're the ones who really need our aid. Programs for the elderly (like Social Security) might count here too.
• meet people's basic human needs, rather than giving them money which they can spend on anything that they want (e.g., SNAP, housing assistance, Medicaid). Everyone ought to be fed; giving people food doesn't feel like you're being taken advantage of in the same way as giving money (and similarly with healthcare, housing, etc.). In some cases, like SNAP (food stamps), this can be pretty close to a pure cash benefit since money is fungible and pretty much everybody buys food. In other cases it involves a lot more meddling.
• reward good behavior, giving money to poor people who are doing something to deserve it - especially those that are working or seeking work (e.g., EITC, TANF, job training). This category has grown the most over the past couple decades, with welfare reform (making welfare dependent on employment-seeking) and large increases to the earned income tax credit. EITC is pretty much what you're looking for - a negative income tax - but it only applies to people who are working.
• are shared benefits for everyone, not special benefits for the poor (e.g., Social Security, Medicare, Alaska's oil fund). The social insurance model fits this category. I paid into Medicare so I deserve my benefits. Anyone who gets hurt on the job and is unable to work will collect disability insurance. That's our oil so we should share the profits. (Alaska's program has the additional advantage of not needing tax money for funding.)

I think that all of the main safety net programs (listed on the sites linked above) fit pretty cleanly within one (or more) of these four categories, except maybe Pell Grants for college expenses (which is at least loosely related to each of them, but might be better thought of as a part of the publicly funded education system).

To fit into one of these four categories, cash transfers would either need to go to everybody (which would be a huge program, e.g. $10,000 per American would require collecting 20% of GDP in taxes and then redistributing it), or only go to the working poor (which is already done with the EITC), or only go to the particularly needy (which is already done for the disabled and the elderly, and wouldn't really work for children). Unconditional cash transfers to generic poor people just don't poll well, and leave politicians vulnerable to political attacks.

Changes at the margin could involve increasing the size of programs that are closest to pure cash transfers (like EITC), shrinking or eliminating programs that involve more meddling (like some kinds of housing assistance), or loosening the restrictions/conditions on specific programs so that there's less hassle in running it (e.g. eligibility requirements for SNAP). Each individual change has an obvious partisan valence. Another option is to try to make the tax system more progressive (or oppose efforts to make it less progressive) - a simple way for the government to get more money into the hands of the poor people is to collect less tax money from them in the first place. That also has an obvious partisan valence.